Integrand size = 13, antiderivative size = 143 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}+\frac {3 b}{2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {3 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}-\frac {3 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2202, 2194, 2191, 2188, 29} \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}+\frac {3 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}-\frac {3 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {3 b}{2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))} \]
[In]
[Out]
Rule 29
Rule 2188
Rule 2191
Rule 2194
Rule 2202
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {(3 b) \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \\ & = \frac {3 b}{2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}-\frac {\left (3 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )} \\ & = -\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}+\frac {3 b}{2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}-\frac {\left (3 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2} \\ & = -\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}+\frac {3 b}{2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {\left (3 b^2\right ) \int \frac {1}{x} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}-\frac {\left (3 b^3\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2} \\ & = -\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}+\frac {3 b}{2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {3 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2} \\ & = -\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}+\frac {3 b}{2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {3 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}-\frac {3 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {2 b^3 x^3-6 b x \coth ^{-1}(\tanh (a+b x))^2+\coth ^{-1}(\tanh (a+b x))^3-3 b^2 x^2 \coth ^{-1}(\tanh (a+b x)) \left (-1+2 \log (x)-2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 x^2 \coth ^{-1}(\tanh (a+b x)) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^4} \]
[In]
[Out]
Timed out.
\[\int \frac {1}{x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}d x\]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {2 \, {\left (48 \, a b^{2} x^{2} + 24 \, a^{2} b x + i \, \pi ^{3} - 6 \, \pi ^{2} {\left (b x - a\right )} - 8 \, a^{3} + 12 i \, \pi {\left (2 \, b^{2} x^{2} + 2 \, a b x - a^{2}\right )} - 24 \, {\left (2 \, b^{3} x^{3} + i \, \pi b^{2} x^{2} + 2 \, a b^{2} x^{2}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right ) + 24 \, {\left (2 \, b^{3} x^{3} + i \, \pi b^{2} x^{2} + 2 \, a b^{2} x^{2}\right )} \log \left (x\right )\right )}}{32 \, a^{4} b x^{3} + i \, \pi ^{5} x^{2} + 32 \, a^{5} x^{2} + 2 \, \pi ^{4} {\left (b x^{3} + 5 \, a x^{2}\right )} - 8 i \, \pi ^{3} {\left (2 \, a b x^{3} + 5 \, a^{2} x^{2}\right )} - 16 \, \pi ^{2} {\left (3 \, a^{2} b x^{3} + 5 \, a^{3} x^{2}\right )} + 16 i \, \pi {\left (4 \, a^{3} b x^{3} + 5 \, a^{4} x^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx=\int \frac {1}{x^{3} \operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {48 \, b^{2} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}} + \frac {48 \, b^{2} \log \left (x\right )}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}} + \frac {2 \, {\left (24 \, b^{2} x^{2} + \pi ^{2} + 4 i \, \pi a - 4 \, a^{2} - 6 \, {\left (i \, \pi b - 2 \, a b\right )} x\right )}}{2 \, {\left (i \, \pi ^{3} b - 6 \, \pi ^{2} a b - 12 i \, \pi a^{2} b + 8 \, a^{3} b\right )} x^{3} + {\left (\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}\right )} x^{2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {48 \, b^{2} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}} + \frac {48 \, b^{2} \log \left (x\right )}{\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}} + \frac {16 \, b^{2}}{-2 i \, \pi ^{3} b x - 12 \, \pi ^{2} a b x + 24 i \, \pi a^{2} b x + 16 \, a^{3} b x + \pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}} - \frac {4 \, {\left (i \, \pi - 8 \, b x + 2 \, a\right )}}{-2 i \, \pi ^{3} x^{2} - 12 \, \pi ^{2} a x^{2} + 24 i \, \pi a^{2} x^{2} + 16 \, a^{3} x^{2}} \]
[In]
[Out]
Time = 7.85 (sec) , antiderivative size = 689, normalized size of antiderivative = 4.82 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {2\,{\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^3-2\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^3-6\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,{\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2+6\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-32\,b^3\,x^3+24\,b\,x\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2+24\,b\,x\,{\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2-24\,b^2\,x^2\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+24\,b^2\,x^2\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-b^2\,x^2\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\mathrm {atan}\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x}\right )\,96{}\mathrm {i}-48\,b\,x\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+b^2\,x^2\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\mathrm {atan}\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x}\right )\,96{}\mathrm {i}}{x^2\,\left (\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,{\left (\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^4} \]
[In]
[Out]